Chapter 6 Exponential and Logarithmic Functions
6.3 Logarithmic Functions
Learning Objectives
In this section, you will:
- Convert from logarithmic to exponential form.
- Convert from exponential to logarithmic form.
- Evaluate logarithms.
- Use common logarithms.
- Use natural logarithms.
In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes.4 One year later, another stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,5 like those shown in Figure 1. Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scale,6 whereas the Japanese earthquake registered a 9.0.7
The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is
Converting from Logarithmic to Exponential Form
In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is
We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve

Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in Figure 2 passes the horizontal line test. The exponential function
We read a logarithmic expression as “The logarithm with base
We can express the relationship between logarithmic form and its corresponding exponential form as follows:
Note that the base
Because logarithm is a function, it is most correctly written as
We can illustrate the notation of logarithms as follows:
Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means
Definition of the Logarithmic Function
A logarithm base
For
- we read
as, “the logarithm with base of ” or the “log base of “ - the logarithm
is the exponent to which must be raised to get
Also, since the logarithmic and exponential functions switch the
- the domain of the logarithm function with base
is . - the range of the logarithm function with base
is .
Can we take the logarithm of a negative number?
No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.
How To
Given an equation in logarithmic form
- Examine the equation
and identify - Rewrite
as
Converting from Logarithmic Form to Exponential Form
Write the following logarithmic equations in exponential form.
Show Solution
First, identify the values of
Here, Therefore, the equation is equivalent to Here, Therefore, the equation is equivalent to
Try It
Write the following logarithmic equations in exponential form.
Show Solution
is equivalent to is equivalent to
Choose the correct exponential form for the following logarithmic equations.
Converting from Exponential to Logarithmic Form
To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base
Converting from Exponential Form to Logarithmic Form
Write the following exponential equations in logarithmic form.
Show Solution
First, identify the values of
Here, and Therefore, the equation is equivalent to Here, and Therefore, the equation is equivalent to Here, and Therefore, the equation is equivalent to
Try It
Write the following exponential equations in logarithmic form.
Show Solution
is equivalent to is equivalent to is equivalent to
Select True if the equations are equivalent:
Select the correct logarithmic form of the equation:
Select True if the equations are equivalent:
Evaluating Logarithms
Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider
Now consider solving
- We ask, “To what exponent must 7 be raised in order to get 49?” We know
Therefore, - We ask, “To what exponent must 3 be raised in order to get 27?” We know
Therefore,
Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate
- We ask, “To what exponent must
be raised in order to get ” We know and so Therefore,
How To
Given a logarithm of the form
- Rewrite the argument
as a power of - Use previous knowledge of powers of
identify by asking, “To what exponent should be raised in order to get ”
Solving Logarithms Mentally
Solve
Show Solution
First we rewrite the logarithm in exponential form:
We know
Therefore,
Try It
Solve
Show Solution
Evaluating the Logarithm of a Reciprocal
Evaluate
Show Solution
First we rewrite the logarithm in exponential form:
We know
Therefore,
Try It
Evaluate
Show Solution
Using Common Logarithms
Sometimes we may see a logarithm written without a base. In this case, we assume that the base is
Definition of the Common Logarithm
A common logarithm is a logarithm with base
For
We read
The logarithm
How To
Given a common logarithm of the form
- Rewrite the argument
as a power of - Use previous knowledge of powers of
to identify by asking, “To what exponent must be raised in order to get ”
Finding the Value of a Common Logarithm Mentally
Evaluate
Show Solution
First we rewrite the logarithm in exponential form:
Therefore,
Try It
Evaluate
Show Solution
How To
Given a common logarithm with the form
- Press [LOG].
- Enter the value given for
followed by [ ) ]. - Press [ENTER].
Finding the Value of a Common Logarithm Using a Calculator
Evaluate
Show Solution
- Press [LOG].
- Enter 321, followed by [ ) ].
- Press [ENTER].
Rounding to four decimal places,
Analysis
Note that
Try It
Evaluate
Show Solution
Rewriting and Solving a Real-World Exponential Model
The amount of energy released from one earthquake was
Show Solution
We begin by rewriting the exponential equation in logarithmic form.
Next we evaluate the logarithm using a calculator:
- Press [LOG].
- Enter
followed by [ ) ]. - Press [ENTER].
- To the nearest thousandth,
The difference in magnitudes was about
Try It
The amount of energy released from one earthquake was
Show Solution
The difference in magnitudes was about
Using Natural Logarithms
The most frequently used base for logarithms is
Most values of
Definition of the Natural Logarithm
A natural logarithm is a logarithm with base
For
We read
The logarithm
Since the functions
How To
Given a natural logarithm with the form
- Press [LN].
- Enter the value given for
followed by [ ) ]. - Press [ENTER].
Evaluating a Natural Logarithm Using a Calculator
Evaluate
Show Solution
- Press [LN].
- Enter
followed by [ ) ]. - Press [ENTER].
Rounding to four decimal places,
Try It
Evaluate
Show Solution
It is not possible to take the logarithm of a negative number in the set of real numbers.
Key Equations
Definition of the logarithmic function | For |
Definition of the common logarithm | For |
Definition of the natural logarithm | For |
Key Concepts
- The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
- Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm.
- Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm.
- Logarithmic functions with base
can be evaluated mentally using previous knowledge of powers of . - Common logarithms can be evaluated mentally using previous knowledge of powers of
- When common logarithms cannot be evaluated mentally, a calculator can be used.
- Real-world exponential problems with base
can be rewritten as a common logarithm and then evaluated using a calculator. - Natural logarithms can be evaluated using a calculator.
Section Exercises
Verbal
- What is a base
logarithm? Discuss the meaning by interpreting each part of the equivalent equations and for
Show Solution
A logarithm is an exponent. Specifically, it is the exponent to which a base
- How is the logarithmic function
related to the exponential function What is the result of composing these two functions?
- How can the logarithmic equation
be solved for using the properties of exponents?
Show Solution
Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation
- Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base
and how does the notation differ?
- Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base
and how does the notation differ?
Show Solution
The natural logarithm is a special case of the logarithm with base
Algebraic
For the following exercises, rewrite each equation in exponential form.
Show Solution
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For the following exercises, rewrite each equation in logarithmic form.
Show Solution
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For the following exercises, solve for
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For the following exercises, use the definition of common and natural logarithms to simplify.
Show Solution
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Numeric
For the following exercises, evaluate the base
Show Solution
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For the following exercises, evaluate the common logarithmic expression without using a calculator.
Show Solution
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For the following exercises, evaluate the natural logarithmic expression without using a calculator.
Show Solution
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Technology
For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.
Show Solution
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Extensions
- Is
in the domain of the function If so, what is the value of the function when Verify the result.
Show Solution
No, the function has no defined value for
- Is
in the range of the function If so, for what value of Verify the result.
- Is there a number
such that If so, what is that number? Verify the result.
Show Solution
Yes. Suppose there exists a real number
- Is the following true:
Verify the result.
- Is the following true:
Verify the result.
Show Solution
No;
Real-World Applications
- The exposure index
for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation where is the “f-stop” setting on the camera, and is the exposure time in seconds. Suppose the f-stop setting is and the desired exposure time is seconds. What will the resulting exposure index be?
- Refer to the previous exercise. Suppose the light meter on a camera indicates an
of and the desired exposure time is 16 seconds. What should the f-stop setting be?
Show Solution
- The intensity levels I of two earthquakes measured on a seismograph can be compared by the formula
where is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0.8 How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.
Footnotes
4http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/#summary. Accessed 3/4/2013.
5http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#summary. Accessed 3/4/2013.
6http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed 3/4/2013.
7http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#details. Accessed 3/4/2013.
8http://earthquake.usgs.gov/earthquakes/world/historical.php. Accessed 3/4/2014.
Glossary
- common logarithm
- the exponent to which 10 must be raised to get
is written simply as
- logarithm
- the exponent to which
must be raised to get written
- natural logarithm
- the exponent to which the number
must be raised to get is written as
Media Attributions
- 6.3 Figure 1 (image) © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
- 6.3 Figure 2 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
- 6.3 Relationship logarithmic form to exponential form (image 1) © OpenStax Precalculus is licensed under a CC BY (Attribution) license
- 6.3 Converting logarithmic form to exponential form (image 2) © OpenStax Precalculus is licensed under a CC BY (Attribution) license